Solvable Groups with π-Isolators

• Published in: Proceedings of the American Mathematical Society 1984-02, Vol.90 (2), p.173-177
• Main Authors: Rhemtulla, A. H., Weiss, A., Yousif, M.
• Format: Article
• Language: English
• Subjects: Mathematical theorems; Mathematics; Prime numbers
• ISSN: 0002-9939; 1088-6826
• DOI: 10.2307/2045332

Let π be any nonempty set of prime numbers. A natural number is a π-number precisely if all of its prime factors are in π. A group G is said to have the π-isolator property if for every subgroup H of G, the set$\root\pi\of H = \{g \in G; g^n \in H,\quad\text{for some}\quad \pi-\text{number}\quad n \}$is a subgroup of G. It is well known that nilpotent groups have the π-isolator property for any nonempty set π of primes. Finitely generated solvable linear groups with finite Prüfer rank, and in particular polycyclic groups, have subgroups of finite index with the π-isolator property if π is the set of all primes. It is shown here that if π is any finite nonempty set of primes and G is a finitely generated solvable group, then G has a subgroup of finite index with the π-isolator property if and only if G is nilpotent-by-finite.